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Homework Help: Limit problem from past exam.

  1. Apr 6, 2010 #1
    limit problem from past exam. :(

    1. The problem statement, all variables and given/known data
    Problem from an exam that I just did is really bugging me.

    lim(x,y)->(0,0) (x*y2)/(x2+y4)






    The attempt at a solution[/b]
    At first I thought the limit didn't exist. I tried using paths, x=y1/2, but then I couldn't find a counter example for which a path approached at a different point. So then I tried using the squeeze theorem, and got that the limit is 0. :\

    Was my approach to the problem correct? is the limit really 0?


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  2. jcsd
  3. Apr 6, 2010 #2

    Dick

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    Re: limit problem from past exam. :(

    You didn't say how you proved it. But yes, the limit is zero.
     
  4. Apr 6, 2010 #3
    Re: limit problem from past exam. :(


    Oh really? XD Well I assumed:

    0<y2<x2+y4 (since x^2 and y^4 are always positive, and y2<y4

    0<y2/x2+y4<1
    Then multiplying x into the inequality:

    0<x*y2/x2+y4<x

    Then taking the limit:
    lim(x,y)->(0,0) 0<lim(x,y)->(0,0) x*y2/x2+y4<lim(x,y)->(0,0) x

    Which gave me that:
    0<lim(x,y)->(0,0) x*y2/x2+y4<0

    which by the squeeze theorem, limit is zero. Was that correct...?
     
  5. Apr 6, 2010 #4

    Dick

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    Re: limit problem from past exam. :(

    Well the starting point 0<y^2<x^2+y^4 is wrong. Suppose x=0 and y=(1/2). I would use polar coordinates.
     
  6. Apr 6, 2010 #5
    Re: limit problem from past exam. :(

    Limit doesn't exist , try y=x^(1/2)
     
  7. Apr 6, 2010 #6
    Re: limit problem from past exam. :(

    Aaah... I was starting it with polar coordinates at first and it seemed like the best way to go, but then it seemed to messy so I thought I was doing it wrong. I was running out of time and just left it like that. XD Thanks lots, at least maybe I have partial credit haha..
     
  8. Apr 6, 2010 #7
    Re: limit problem from past exam. :(

    lim x->0 f(x,x^(1/2))=1/2
    and lim x->0 f(x,0)=0
     
  9. Apr 6, 2010 #8

    Dick

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    Re: limit problem from past exam. :(

    Ooops. You are so right. Sorry.
     
  10. Apr 6, 2010 #9
    Re: limit problem from past exam. :(

    Aww darn... it was a 3 part question worth 5 points total on that part of the exam, so I should have 10/3 credit hehe... The rest of the test was pretty good.
     
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